Randomized Polynomial-Time Root Counting in Prime Power Rings

Suppose \(k,p\!\in\!\mathbb{N}\) with \(p\) prime and \(f\!\in\!\mathbb{Z}[x]\) is a univariate polynomial with degree \(d\) and all coefficients having absolute value less than \(p^k\). We give a Las Vegas randomized algorithm that computes the number of roots of \(f\) in \(\mathbb{Z}/\!\left(p^k\right)\) within time \(d^3(k\log p)^{2+o(1)}\). (We in fact prove a more intricate complexity bound that is slightly better.) The best previous general algorithm had (deterministic) complexity exponential in \(k\). We also present some experimental data evincing the potential practicality of our algorithm.